In this paper we study the global existence of positive integrable solution for the nonlinear integral inclusion of fractional order \[ x(t) \in p(t) + I^\alpha F_1 (t, I^\beta f_2 (t, x(\varphi(t)))),\quad t \in (0, 1). \] As an application the global existence of the solution for the initial-value problem of the arbitrary (fractional) orders differential inclusion \[ \frac{dx(t)}{dt}\in p(t)+ I^\alphaF_1(t,D^\gammax(t))),\quad \text{a.e.}\ t gt 0 \] will be studied.
In this paper we use the upper and lower solutions method combined with Schauder's fixed point theorem and a fixed point theorem for condensing multivalued maps due to Martelli to investigate the existence of solutions for some classes of partial Hadamard fractional integral equations and inclusions.
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